551 5.3 Trigonometric Function Values and Angle Measures Trigonometric Function Values of Special Angles Certain special angles, such as 30°, 45°, and 60°, occur so often in trigonometry and in more advanced mathematics that they deserve special study. We start with an equilateral triangle, a triangle with all sides of equal length. Each angle of such a triangle measures 60°. Although the results we will obtain are independent of the length, for convenience we choose the length of each side to be 2 units. See Figure 30(a). Bisecting one angle of this equilateral triangle leads to two right triangles, each of which has angles of 30°, 60°, and 90°, as shown in Figure 30(b). An angle bisector of an equilateral triangle also bisects the opposite side. Thus the shorter leg has length 1. Let x represent the length of the longer leg. 22 = 12 + x2 Pythagorean theorem 4 = 1 + x2 Apply the exponents. 3 = x2 Subtract 1 from each side. 2 3 = x Square root property; Choose the positive root. Figure 31 summarizes our results using a 30°960° right triangle. As shown in the figure, the side opposite the 30° angle has length 1. For the 30° angle, hypotenuse = 2, side opposite = 1, side adjacent = 23. Now we use the definitions of the trigonometric functions. sin 30° = side opposite hypotenuse = 1 2 cos 30° = side adjacent hypotenuse = 3 2 tan 30° = side opposite side adjacent = 12 3 = 12 3 # 23 23 = 23 3 csc 30° = 2 1 = 2 sec 30° = 22 3 = 22 3 # 23 23 = 223 3 cot 30° = 23 1 = 23 608 2 308 1 Á3 Figure 31 EXAMPLE 3 FindingTrigonometric Function Values for 60° Find the six trigonometric function values for a 60° angle. SOLUTION Refer to Figure 31 to find the following ratios. sin 60° = 23 2 cos 60° = 1 2 tan 60° = 23 1 = 23 csc 60° = 22 3 = 223 3 sec 60° = 2 1 = 2 cot 60° = 12 3 = 23 3 S Now Try Exercises 31, 33, and 35. 608 608 608 2 2 2 Equilateral triangle (a) 608 308 608 1 2 2 308 1 908 908 x x 308–608 right triangle (b) Figure 30 Rationalize the denominators.
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